The effect of travel restrictions on the spread of a moderately contagious disease

Transcription

1 The effect of travel restrctons on the spread of a moderately contagous dsease Martn Camtz 1,2,3,4 Fredrk Lljeros 3 1 Swedsh Insttute for Infectous Dsease Control, Solna, Sweden 2 Department of Medcal Epdemology and Bostatstcs, Karolnska Insttute, Solna, Sweden 3 Department of Socology, Stockholm Unversty, Stockholm, Sweden 4 Theoretcal Bologcal Physcs, Department of Physcs, Royal Insttute of Technology, Stockholm, Sweden Correspondng author Emal addresses: MC: martn.camtz@sm.k.se FL: fredrk.lljeros@socology.su.se

2 Abstract Background Much research n epdemology has been focused on evaluatng conventonal methods of control strateges n the event of an epdemc or pandemc. Travel restrctons are often suggested as an effcent way to reduce the spread of a contagous dsease that threatens publc health, but few papers have studed n depth the effects of travel restrctons. In ths study, we nvestgated what effect dfferent levels of travel restrctons mght have on the speed and geographcal spread of an outbreak of a dsease smlar to severe acute respratory syndrome (SARS). Methods We used a stochastc smulaton model ncorporatng survey data of travel patterns between muncpaltes n Sweden collected over 3 years. We tested scenaros of travel restrctons n whch travel over dstances > 50 km and 20 km would be banned, takng nto account dfferent levels of complance. Results We found that a ban on journeys > 50 km would drastcally reduce the speed and geographcal spread of outbreaks, even when complance s < 100%. The result was found to be robust for dfferent rates of ntermuncpalty transmsson ntenstes. Conclusons Ths study supports travel restrctons as an effectve way to mtgate the effect of a future dsease outbreak.

3 Background Knowledge of the speed at whch a contagous dsease travels between geographcal regons s vtal for makng decsons about the most effectve nterventon strateges. The actual routes a dsease wll take are strongly determned by how ndvduals travel wthn and between regons [1-4]. As was shown durng the outbreak of severe acute respratory syndrome (SARS) [5], current travel patterns enable contagous dseases to spread to far corners of the globe at alarmng rates. Ths demonstrates the need for a new type of model that ncorporates travel networks. Several authors have responded to the call, resultng n now-classc papers. Rvachev and Longn [6], wth ther followers Gras et al [7], were among the frst to publsh such studes, usng determnstc models. Hufnagel et al [8] have demonstrated how a smple stochastc model n conjuncton wth data on avaton traffc could be used to smulate the global spread of the SARS epdemc. Usng a stochastc transmsson model on both a cty level and globally, wth each cty nterconnected by the nternatonal avaton network, they produced results n surprsng agreement wth World Health Organzaton (WHO) reports of the actual epdemc. Wth some exceptons, most promnently Japan, all nfected countres n the smulaton were also present n the WHO reports. The orders of magntude were also closely matched. Our study appled a verson of the Hufnagel model to Sweden n order to predct the effect that travel restrctons mght have on the geographcal spread of an outbreak. Instead of usng only the avaton network, whch connects only some 30 towns n Sweden, we used survey data on all ntermuncpal travel, ncludng all forms of travel. Sweden s, by European standards, a large country, wth a small populaton. Just over 9 mllon people share square klometers. The populaton s, however, largely urbanzed, and n that respect smlar to other ndustralzed natons wth large areas. Eubank et al [9] estmated a travel network at communty level usng census data. Our data drectly cover travelng over the whole naton on all scales, although we have kept only the regonal data. It s suffcently extensve that smulaton or smoothng for estmatng a travel network could be avoded and thus can, n some respects, be regarded as real. Usng such data n a smulaton at ths geographc level, s unque. The choce of a stochastc modelng approach [10] was based on the fact that t mmcs the hghly random ntal phase of an epdemc better than does the tradtonal determnstc approach[11, 12]. We frst present the survey data used to estmate travel ntenstes between dfferent muncpaltes n Sweden. We then ntroduce the smulaton model for smulatng the spread of the dseases and study the effect of travel restrctons. Ths ntroducton s to some extents a recaptulaton of Hufnagel s model. Followng ths, we present the results of the smulatons. We conclude our

4 study wth a dscusson of the valdty of the model and possble conclusons for future polcy nterventons. Methods For ths study, we used data from a random survey carred out by Statstcs Sweden from 1999 to 2001, nclusve [13]. A total of ndvduals took part n the survey, consttutng 71.9% of the selecton. In all, dstnct ntermuncpal trps were reported. An ntermuncpal journey was defned as a trp between two ponts where the ndvdual lves, works, or conducts an errand. In other words, we treated a journey between home and work as several trps f the traveler made stops on the way for errands, provded that a muncpal border was crossed between each stop. The data were weghted to correspond to 1 day and to the entre populaton for ages 6 to 84 years. For a more detaled descrpton of the travel data, see Appendx A n the supplementary materal. As t turned out, roughly 1% of the data was sgnfcantly erroneous and was consequently removed *. From the remanng set, we estmated a travel ntensty matrx wth each element correspondng to the one-way travel ntensty between two muncpaltes. The number of non-zero elements was (to be compared wth the sze of the matrx: ). The matrx elements stood n drect correspondence wth the underlyng data, weghted for tme and populaton. Even though the matrx gves a good pcture of the travelng pattern n Sweden, we must treat any travel ntensty between two specfc communtes wth care. Ths s especally true for small communtes wth only a sngle or very few journeys made between them. A total of nne scenaros, wth 1000 realzatons each, was smulated to study the effects of three levels of travel restrctons as a control measure, for three dfferent levels of the global ntercommunty nfectousness parameter, γ, whch was used to calbrate the model n the study of Hufnagel et al. Sxty days was chosen as the smulaton perod, as ths gves suffcent tme for a possble extncton to occur and for all stochastcty to play out ts part n all but the smallest and most dstant muncpaltes. Each scenaro started wth a sngle nfectous ndvdual n Stockholm, and treated the country as solated from nflux of dsease. The travelng restrctons were dvded nto three levels. In the frst level, we used the complete ntensty matrx. In the followng two, we removed data correspondng to journeys > 50 km and journeys > 20 km, respectvely. The smulatons were desgnated SIM, SIM50, and SIM20, respectvely. In Fgure 1, the datasets are dsplayed as geographcal plots. We also consdered the case f the travel restrctons were not obeyed wholly by the publc. Perhaps 5% mght not heed the restrctons, resultng n a small but non-zero ntensty for trps longer than the set restrctons. Full 1000-run smulatons were * The erroneous records were long-dstance journeys, mostly between ndvdual communtes n an unreasonably short tme. Had they not been removed, ther nfluence would have been sgnfcant, acceleratng the spread across the country. The correct data were rretrevable but the effect of ther absence was deemed wthn the margn of error for long-dstance journeys.

5 made at varyng levels of dstance restrctons and complance, resultng n a mesh surface of the ncdence. We used a smplfed verson of the model suggested by Hufnagel et al [8], and thus the followng s as much a descrpton of Hufnagel s model as our own. The ndvduals n both models can be n four dfferent states: S: susceptble L: latent, meanng nfected but not nfectous I: nfectous R: recovered and/or mmune. The rate at whch ndvduals move from one category to the next s governed by the ntensty parameters: α=0.55 ; β=0.21, whch s the nverse nfectous tme and ν=0.19, the nverse latency tme[14, 15]. Indvduals become nfected at a rate proportonal to α and the number of nfected (force of nfecton). They subsequently become nfectous at rate β, contrbutng to the force of nfecton, and recover at rate ν. Thus far, ths s a descrpton of a regular random-mxng epdemc model. We now assume a travelng component contrbutng to the force of nfecton, a term for each of the connected muncpaltes proportonal to the number of nfectous there. As the process s assumed to be Markovan, as n Hufnagel s model, the tme between two events, t, s random, taken from an exponental dstrbuton, ( Q) t Exp 1 (1) where Q s the total ntensty, the sum of all ndependent transmsson rates: S S = α + I j, N L Q I γ j, N j I Q = vl Q = Q R, = βi. L I Q + Q + Q R. (2) These are the equatons that govern the smulatons and gve us the contnuous tme settng. The component γ j, n (2) (note the reversed ndexes) s the ntermuncpal nfectousness correspondng to the one-way route j to. If ω = M, j, M j, j, where M j, s the travel ntensty (.e. ωj, s the probablty that a traveler n j wll choose the route j to, then γ j, =γω j,. In cases where restrctons are actve, ths expresson s further scaled row-wse to match the smaller mass of the matrx. γ s the Some authors refer to ths as the attack rate although ths s not the commonest defnton.

6 global ntermuncpal nfectousness parameter mentoned above. We use the approxmaton gven by Hufnagel et al based on data from the actual outbreak, γ=0.27. The parameter γ s nfluenced by the total travel ntensty, the medum of travel and as we have seen, the propensty for travel n dfferent communtes. We would have lked to calbrate our model n a smlar way, but as we have no outbreak data for Sweden, we needed to see whether changes n γ would drastcally alter our conclusons. To get an dea of ts effect, we compared Hufnagel s estmate of γ wth other possble values. As γ s an nfectousness parameter playng a smlar role n the governng equatons above as α, we argue that α=0.55 s an upper bound for γ. The force of nfecton between two muncpaltes wth equal populaton and number of nfectous s unlkely to be hgher than from wthn those muncpaltes. To fnd an approprate lower bound we extrapolated proportonately from these, producng Although ths s not mentoned n the orgnal work by Hufnagel et al, the expresson above means that everybody, regardless of where they lve, s equally prone to travel outsde ther home, the uptake area of the arport or, n our case, the muncpalty. Ths s a heavy assumpton ndeed, as t depends on the functon of the muncpalty varyng. The muncpalty may be a suburb or self-suffcent communty, just as arports may be transt hubs or termnals. One of the strengths of Hufnagel s model s that t seems to be forgvng towards many smplfcatons, ths one ncluded, wth the correct choce of γ. We nvestgated correctons for ths assumpton, such as row-wse scalng accordng to the known probablty for travel, but found lttle effect on absolute ncdence and none on the qualtatve conclusons of the current study. As such, we were reluctant to stray from Hufnagel s model. After the ntal condtons were set up, ncludng a sngle nfected person n Stockholm, the smulaton ran as follows. Frst, we moved forward n tme wth a random step t gven by (1). We then selected the event that would occur wth a probablty proportonal to the correspondng ntensty. All ntenstes were updated accordng to the new state, and the process was repeated untl the dsease ded out or the smulaton perod, 60 days, was passed. Results The results for all nne scenaros were plotted geographcally and color-coded accordng to the mean ncdence (Fgure 2). A scenaro wth no restrctons resulted n an outbreak n whch a majorty of the muncpaltes became affected regardless of γ. Only the ncdence dffered. A ban on journeys > 50 km stfled the dynamcs of the outbreak. For the two lower values of γ, we see that the dsease remaned n the Stockholm area after 60 days, and for the hgher value of γ, the dsease dd not manage to spread far from the densely populated areas around the largest Swedsh ctes. Prohbtng journeys > 20 km would result n an even slower spread wth a small number of afflcted muncpaltes, manly

7 localzed around Stockholm. What s more, the total ncdence after 60 days as well as the ncdence n each muncpalty dropped as we mposed the restrctons. Table 1 compares the country s total ncdence n the three smulatons for whch Hufnagel s estmate of γ was used. Table 2 presents the ncdence broken down nto a few selected muncpaltes. The reason for the decrease n ncdence s of course the lmted transmsson paths avalable to the dsease. The dsease, after havng spread from one muncpalty to another wll constantly be transmtted back nto the orgnatng muncpalty, provded that there s a flow of travelers n the opposte drecton n the travel ntensty matrx. Travel restrctons lmt both spread to other muncpaltes and rentroducton. For comparson, f traffc s removed altogether, the mean ncdence n Stockholm wll be 917. The process outlned above s also responsble for the decreased number of extncton runs. For a regular contnuous tme branchng process where the number new cases s completely ndependent between ndvduals, one would expect the probablty of extncton to be I/R 0 =37%. and ths was confrmed n the course of testng our model. It s also clear how travel restrctons confer ncreasng protecton on ctes that are further from the captal, the focal pont of the nfecton. The major ctes of Göteborg (Gothenburg) and Malmö would be protected even though traffc nto these ctes s heavy. In fact, the farthest the dsease woudl ever make t n SIM50 s Ljungby, 1471 km from Stockholm and stll some 200 km from Malmö. For SIM20, the farthest cty s Uddevalla, 441 km away and a suburb of Göteborg. The mean reach of the epdemc n those cases s only 276 and 34 km, respectvely. An objecton to the applcablty of ths model s that n all probablty, complete enforcement of the restrctons may not be achevable or even desrable, as n the case of hgh-prorty professonals wth crucal functons n socety durng a crss stuaton. Incdence does ndeed clmb the more restrctons are gnored, but not to such an extent as to render the travel restrctons dubous as a means of dsease control (Fgure 3). A plot wth unrestrcted travel, duplcated from Fgure 2, s gven for comparson. Fgure 4 shows a fner spaced mesh of ncdence versus restrcton dstance and complance. Bear n mnd that there was no attempt to correlate the randomness between the smulaton sets. Therefore, the random numbers used n each were completely ndependent, gvng rse to consderable smulaton nose. Even though the landscape s rough, the trend n both dmensons was clearly vsble. Looser travel restrctons and lower complance means hgher ncdence. Dscusson Our results show clearly that travelng restrctons would have a sgnfcant benefcal effect, reducng both the geographcal spread and the total and local ncdence. Ths holds true for all three levels of ntercommunty nfectousness, γ, smulated, γ s nfluenced by many factors, most notably by total travel ntensty, but also by the

8 medum of travel, the behavor of the traveler, the model of dspersal by travel and the nfectousness of the dsease. Hufnagel et al calbrated γ usng data from the actual outbreak. As mentoned, no attempt was made on our part to fnd the true value of γ n the new settngs, as no such outbreak data are avalable for Sweden. Ths would be consdered a flaw for a quanttatve study on a SARS outbreak n Sweden. By smulatng for dfferent values of the parameter, however, we can be confdent n the qualtatve concluson, namely, that the same general behavor can be expected n the unrestrcted scenaro and n response to the control measures, regardless of γ. The same reasonng supports generalzaton of the results to other countres or regons. The survey travel data gve a farly accurate pcture of travel patterns n Sweden and mrror many western countres. Some countres, e.g. the USA, are more dependent on motor vehcles for commuter traffc, and nfectousness would therefore be antcpated to be lower. Such effects are, agan, ncluded n calbraton of γ. Keep n mnd that the used value of γ was taken from a model ncludng only global ar traffc. Fgure 2 shows that travel restrctons have a postve effect for γ n the proposed range. In lght of the fact that ntermuncpal travel heavly nfluences ncdence even at a local level, we may justfably be concerned about boundary condtons. We treated Sweden as an solated country, but qute obvously, the ncdence wll be underestmated for areas wth frequent traffc across the borders. Ths ncludes n partcular the Öresund regon around Malmö, and to a lesser extent, nternatonal arports and the small towns borderng Norway and Fnland. We would lke to pont out that, as n most epdemologc smulatons, ndvduals are not explctly represented n the model. Ths s also true for ndvduals who are travelng. In realty, people who travel run an ncreased rsk of contractng the dsease. Ths s correctly modeled, as ndvduals who are travelng are ncluded n the travel nflux nto the muncpaltes. The nflux n turn affects the probablty of addtonal nfectons at any gven tme. Of course, t s hghly probable that ths would be the travelers themselves, as, almost wthout excepton, they return to the orgn of ther journey. Even though there s presently no treatment or vaccne for SARS, results show that lmted quarantne as suggested here drastcally decreases the rsk of transmsson, and ths may well turn out to be the most expedent form of nterventon. In many countres, Sweden ncluded, lmtng freedom of travel s unconsttutonal and must take the form of general recommendatons. Addtonally, certan professons of crucal mportance to socety durng a crss stuaton must be exempt from travel restrctons. The study shows that even f a substantal fracton of the populaton breaks the restrctons, ths strategy s stll vable. For other types of dsease for whch preventve treatment (pandemc flu) or vaccne (smallpox) are avalable, our results show that long-dstance travelers are an mportant group for targeted control measures.

9 It s worth notng on the travel ntensty matrx, where the elements drectly reflect the underlyng survey data, that there are several proposed alternatves, usng smoothng technques or data-generatng smulaton. Completng the matrx n such a way would correctly ntroduce many connectons that are mssng from our data, but a substantal number would be falsely represented, and could endanger the valdty of the model due to unforeseen stochastc mechansms. The methods all have nherent mprecsons and flaws, whch unfortunately n ths context would be dffcult to estmate. The choce of one n preference over another would certanly be contended. Our scheme of drect extrapolaton from the raw data s certanly no better but does have the beneft of transparency and reasonable control over errors. As s explaned n further detal n the appendx, ths means that certan connectons between muncpaltes that are used n realty, however nfrequently, are mssng, whle on the other hand some wll be heavly overestmated. Ths s especally true for certan unusual muncpaltes. The routes between the more populated communtes and other heavy connectons are much better estmated, as crude statstcal analyss wll ndcate. Also close to the true value s the travel ntensty as a whole, as well as the summed nflux and outflux of any muncpalty. Conclusons Our methods show that restrctng travel between muncpaltes n such a way that travel above a certan dstance s banned, would ndeed have a benefcal effect on the speed of transmsson of a hghly contagous dsease, geographcally and n absolute numbers. Ths concluson s true for a range of plausble values of the ntermuncpal nfectousness. Even n scenaros of complance as low as 70%, travel restrctons are effectve. Thus, the effectveness of travel restrctons as a means of mtgatng a future epdemc s supported. The model and results are robust and there s no reason to beleve that the results are not generally applcable to any country or regon.

10 Competng nterests The authors have receved fnancal support from the organzatons mentoned n acknowledgements. Authors' contrbutons MC performed all codng and smulatons, carred out analyses and s the man author of the manuscrpt. FL conceved the project and desgn and ntated the work. He partcpated n analyses and draftng of the paper. FL approved the fnal verson of the paper. Acknowledgements Ths study was supported by The Swedsh Insttute for Infectous Dseases Control, Swedsh Councl for Workng Lfe and Socal Research, and the European Unon Research NEST Project (DYSONET ). The authors would lke to express ther grattude to Tom Brtton and Åke Svensson, Department of Mathematcs, Stockhholm Unversty, Monca Nordvk, Department of Socology, Stockholm Unversty and Alden Klovdahl, Socal Scences, Australan Natonal Unversty and the members of S-GEM, Stockholm Group of Epdemc Modelng ( for ther knd support.

11 References 1. Keelng MJ, Woolhouse MEJ, Shaw DJ, Matthews L, Chase-Toppng M, Haydon DT, Cornell SJ, Kappey J, Wlesmth J, Grenfell BT: Dynamcs of the 2001 UK foot and mouth epdemc: Stochastc dspersal n a heterogeneous landscape. Scence 2001, 294(5543): Wyle JL, Jolly A: Patterns of chlamyda and gonorrhea nfecton n sexual networks n Mantoba, Canada. Sexually Transmtted Dseases 2001, 28(1): Grenfell BT, Bjornstad ON, Kappey J: Travellng waves and spatal herarches n measles epdemcs. Nature 2001, 414(6865): Sattenspel L, Detz K: A Structured Epdemc Model Incorporatng Geographc-Moblty among Regons. Mathematcal Boscences 1995, 128(1-2): Pearson H: SARS - What have we learned? Nature 2003, 424(6945): Rvachev LA, Longn IM: A mathematcal-model for the global spread of nfluenza. Mathematcal Boscences 1985, 75(1): Gras RF, Ells JH, Glass GE: Assessng the mpact of arlne travel on the geographc spread of pandemc nfluenza. Eur J Epdemol 2003, 18(11): Hufnagel L, Brockmann D, Gesel T: Forecast and control of epdemcs n a globalzed world. Proceedngs of the Natonal Academy of Scences of the Unted States of Amerca 2004, 101(42): Eubank S, Guclu H, Kumar VS, Marathe MV, Srnvasan A, Toroczka Z, Wang N: Modellng dsease outbreaks n realstc urban socal networks. Nature 2004, 429(6988): Andersson H, Brtton P: Stochastc Epdemc Models and Ther Statstcal Analyss. Volume 151. New York: Sprnger-Verlag; Anderson RM, May RM, Anderson B: Infectous Dseases of Humans: Dynamcs and Control Oxford Unversty Press; Dekmann O, Heesterbeek JAP: Mathematcal Epdemology of Infectous Dseases: Model Buldng, Analyss and Interpretaton. Chchester: John Wley and Sons Ltd.; Statens Insttut för Kommunkatonsanalys SSC: RES Den natonella reseundersöknngen

12 14. Lpstch M, Cohen T, Cooper B, Robns JM, Ma S, James L, Gopalakrshna G, Chew SK, Tan CC, Samore MH et al: Transmsson dynamcs and control of severe acute respratory syndrome. Scence 2003, 300(5627): Ferguson NM, Cummngs DAT, Cauchemez S, Fraser C, Rley S, Meeya A, Iamsrthaworn S, Burke DS: Strateges for contanng an emergng nfluenza pandemc n Southeast Asa. Nature 2005, 437(7056):

13 Fgures Fgure 1 The ntermuncpal travel network The ntermuncpal travel network wth travel ntenstes ndcated by color lnes. The scale s logarthmc n trps per day. SIM shows the complete dataset. In SIM50 and SIM20, all journeys > 50 km and 20 km, respectvely, have been removed. The lnes are drawn between the populaton centers of each muncpalty, so n many cases the trps are shorter than the lnes representng them. Fgure 2 Epdemc spread for dfferent restrctons and values of γ Geographcal plot of the muncpaltes, logarthmcally color-coded accordng to the mean ncdence after 60 days. SIM depcts the complete data set. In SIM50 and SIM20, all journeys > 50 km and 20km, respectvely, have been removed. The red crcle sgnfes the mean extent of the epdemc from Stockholm. Fgure 3 Epdemc spread for dfferent restrctons and complance Geographcal dstrbuton of the ncdence after 60 days shown for SIM50 and SIM20 for dfferent levels of complance. The left plot shows the unrestrcted case wth Hufnagels orgnal γ value for comparson. Ths plot reflects the same data as that on the mddle row, rght column of Fgure 2 but wth scale to match the current fgure. Fgure 4 Total ncdence for varyng complance and restrctons. A surface plot showng ncdence after 60 days wth the parameters of complance and dstance restrctons on the data axes realzatons were made for each pont. The surface has ts hghest values at hgh set dstance lmt and low complance. Its low values are found at opposte corner.

14 Tables Table 1 Man results SIM SIM50 SIM20 Results Mean 95% SI Mean 95% SI Mean 95% SI Total number of nfected Percentage of populaton Intermuncpal nfectons (n) Incdence after 60 days (n) Percentage of populaton , Afflcted muncpaltes (n) Mean ncdence n muncpaltes (n) Mean nfluence dstance (km) Travel ntensty matrx Value Value Value Total travel ntensty (mllons/day) Intermuncpal one-way routes (n) Summary Value Value Value Extncton runs (n) Mean tme for extncton (days) Mean number of afflcted muncpaltes before extncton (n) Total number of realzatons The table shows the man results along wth mscellaneous nformaton about the smulaton. Fgures refer to smulated values at the end of the run, 60 days. The mean, where applcable, was taken over the set of runs that ran ther course through the full 60 days. The extncton runs hence dd not affect the means but ther numbers are of course nterestng n ther own rght. The 95% smulaton ntervals (SI) were calculated by bootstrappng samples. By ncdence, we mean the number of nfectous people. Intermuncpal nfectons s the percentage of the total number of nfected that caught the dsease va ntermuncpal nfecton. There are 289 muncpaltes n Sweden and the populaton s approxmately 8.9 mllon.

15 Table 2 Muncpaltes of key nterest SIM SIM50 SIM20 Muncpalty Mean 95% SI Mean 95% SI Mean 95% SI Stockholm Göteborg Malmö Huddnge Upplands-Bro Norrtälje Södertälje Västerås Esklstuna Umeå Luleå Örebro Jönköpng Lnköpng Helsngborg Borås Gävle Ljungby Hofors Örkelljunga 4.9 3,8 5,0 A selecton of muncpaltes wth the mean ncdence and bootstrapped 95 % smulaton ntervals wth extncton runs fltered out. Ths set and ts orderng s the same for the ndvdual rows n the table, whch explans the zero-valued lower nterval bounds and other dscrepances. After Stockholm, Göteborg and Malmö are the largest ctes n Sweden. The sngle most traveled route s that between Stockholm and neghborng Huddnge, traveled by approxmately people daly, each way. The declne n ncdence closely follows that n Stockholm. Upplands-Bro s representatve of an outer suburb to Stockholm. Södertälje and Norrtälje are nearby towns but are not consdered suburbs. Västerås and Esklstuna are more dstant, but have a far number of commuters. Örebro through Luleå are larger towns at some dstance from Stockholm wth no notable commuter traffc. Fnally, the last four are small towns n southern Sweden. SI, smulaton ntervals

16 Addtonal fles Sample anmatons The anmatons n wmf format provded wth ths manuscrpt show dfferent sngle realzatons or possble scenaro of an unrestrcted epdemc where one ntal nfected person s located n Stockholm. They are provded for nterest only and should not be seen as contrbutng results, whch requre the combned analyss of many more realzatons. Addtonal fle 1 The frst fle, Stockholm.wmf, shows a common scenaro of spread throughout the country from Stockholm. Addtonal fle 2 Uppsala.wmf clearly demonstrates the benefts of stochastc modelng n that the epcenter s spontaneously translocated to Uppsala, north of Stockholm, and the epdemc s hence delayed. Ths s qute a plausble event that would not be captured n determnstc models. Addtonal fle 3 Des_out.wmf shows the common event of the epdemc not catchng on but rather dyng out wthn a few days. All these realzatons have been aborted after 60 days. Addtonal fle 4 Bggy.wmf s smlar to the frst but was not aborted untl the epdemc has peaked and ded out due to depleton of susceptble ndvduals. Supplementary materal Appendx A Ths appendx descrbes the travel survey data from Statstcs Sweden n greater detal.

17 SIM SIM50 SIM Fgure 1 10

18 SIM SIM50 SIM γ= γ= γ=.13 1 Fgure 2

19 100% 99% 95% 90% 70% SIM Unrestrcted 100 SIM Fgure 3

20 x Incdence Fgure 4 80 Complance (%) Dstance restrcton (km) 50

21 Addtonal fles provded wth ths submsson: Addtonal fle 5 : camtz sup5-copy edted author revewed 2.doc : 47Kb Addtonal fle 4 : Bggy.wmv : 3628Kb Addtonal fle 3 : Des_out.wmv : 42Kb Addtonal fle 2 : Uppsala.wmv : 1003Kb Addtonal fle 1 : Stockholm.wmv : 1034Kb